Theorem apsym 8937

Description: Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)

Assertion

Ref	Expression
apsym	|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )

Proof of Theorem apsym

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8323	. . 3 |- ( B e. CC -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
2	1	adantl 277	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) )
3		cnre 8323	. . . . . 6 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
4	3	ad3antrrr 496	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
5		simplrl 541	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> x e. RR )
6		simplrl 541	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> z e. RR )
7	6	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> z e. RR )
8		reaplt 8919	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR ) -> ( x # z <-> ( x < z \/ z < x ) ) )
9	5, 7, 8	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> ( x < z \/ z < x ) ) )
10		reaplt 8919	. . . . . . . . . . . . 13 |- ( ( z e. RR /\ x e. RR ) -> ( z # x <-> ( z < x \/ x < z ) ) )
11	7, 5, 10	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z # x <-> ( z < x \/ x < z ) ) )
12		orcom 740	. . . . . . . . . . . 12 |- ( ( x < z \/ z < x ) <-> ( z < x \/ x < z ) )
13	11, 12	bitr4di 198	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( z # x <-> ( x < z \/ z < x ) ) )
14	9, 13	bitr4d 191	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( x # z <-> z # x ) )
15		simplrr 542	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> y e. RR )
16		simplrr 542	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> w e. RR )
17	16	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> w e. RR )
18		reaplt 8919	. . . . . . . . . . . 12 |- ( ( y e. RR /\ w e. RR ) -> ( y # w <-> ( y < w \/ w < y ) ) )
19	15, 17, 18	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> ( y < w \/ w < y ) ) )
20		reaplt 8919	. . . . . . . . . . . . 13 |- ( ( w e. RR /\ y e. RR ) -> ( w # y <-> ( w < y \/ y < w ) ) )
21	17, 15, 20	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w # y <-> ( w < y \/ y < w ) ) )
22		orcom 740	. . . . . . . . . . . 12 |- ( ( y < w \/ w < y ) <-> ( w < y \/ y < w ) )
23	21, 22	bitr4di 198	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( w # y <-> ( y < w \/ w < y ) ) )
24	19, 23	bitr4d 191	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( y # w <-> w # y ) )
25	14, 24	orbi12d 805	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x # z \/ y # w ) <-> ( z # x \/ w # y ) ) )
26		apreim 8934	. . . . . . . . . 10 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
27	5, 15, 7, 17, 26	syl22anc 1279	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( x # z \/ y # w ) ) )
28		apreim 8934	. . . . . . . . . 10 |- ( ( ( z e. RR /\ w e. RR ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) <-> ( z # x \/ w # y ) ) )
29	7, 17, 5, 15, 28	syl22anc 1279	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) <-> ( z # x \/ w # y ) ) )
30	25, 27, 29	3bitr4d 220	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) <-> ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) ) )
31		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> A = ( x + ( _i x. y ) ) )
32		simpllr 540	. . . . . . . . 9 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> B = ( z + ( _i x. w ) ) )
33	31, 32	breq12d 4143	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> ( x + ( _i x. y ) ) # ( z + ( _i x. w ) ) ) )
34	32, 31	breq12d 4143	. . . . . . . 8 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( B # A <-> ( z + ( _i x. w ) ) # ( x + ( _i x. y ) ) ) )
35	30, 33, 34	3bitr4d 220	. . . . . . 7 |- ( ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) /\ A = ( x + ( _i x. y ) ) ) -> ( A # B <-> B # A ) )
36	35	ex 115	. . . . . 6 |- ( ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( A = ( x + ( _i x. y ) ) -> ( A # B <-> B # A ) ) )
37	36	rexlimdvva 2676	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A # B <-> B # A ) ) )
38	4, 37	mpd 13	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) /\ B = ( z + ( _i x. w ) ) ) -> ( A # B <-> B # A ) )
39	38	ex 115	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( z e. RR /\ w e. RR ) ) -> ( B = ( z + ( _i x. w ) ) -> ( A # B <-> B # A ) ) )
40	39	rexlimdvva 2676	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( E. z e. RR E. w e. RR B = ( z + ( _i x. w ) ) -> ( A # B <-> B # A ) ) )
41	2, 40	mpd 13	1 |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   CCcc 8178   RRcr 8179   _ici 8182   + caddc 8183   x. cmul 8185   < clt 8361   # cap 8912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913

This theorem is used by:  addext  8941  mulext  8945  ltapii  8966  ltapd  8969  aptap  8981  apdivmuld  9146  div2subap  9170  recgt0  9183  prodgt0  9185  irrmulap  10059  pwm1geoserap1  12294  absgtap  12296  geolim  12297  geolim2  12298  geo2sum2  12301  geoisum1c  12306  tanaddap  12525  egt2lt3  12566  sqrt2irraplemnn  12978  logdivlt  16088  1sgm2ppw  16250  triap  17244  apdiff  17264